Light Affine Logic (Proof Nets, Programming Notation, P-Time Correctness and Completeness)

TL;DR

This paper introduces Light Affine Logic, demonstrating its polynomial-time cut elimination and encoding of all polynomial-time Turing machines, with proofs of correctness and completeness, and illustrating programming within resource constraints.

Contribution

It provides the first structured presentation of Light Affine Logic, proving P-Time correctness and completeness, and offering a compact programming notation for resource-limited computation.

Findings

Light Affine Logic has polynomially bounded cut elimination.

It encodes all polynomial-time Turing machines.

The paper includes programming examples demonstrating resource-limited computation.

Abstract

This paper is a structured introduction to Light Affine Logic, and to its intuitionistic fragment. Light Affine Logic has a polynomially costing cut elimination (P-Time correctness), and encodes all P-Time Turing machines (P-Time completeness). P-Time correctness is proved by introducing the Proof nets for Intuitionistic Light Affine Logic. P-Time completeness is demonstrated in full details thanks to a very compact program notation. On one side, the proof of P-Time correctness describes how the complexity of cut elimination is controlled, thanks to a suitable cut elimination strategy that exploits structural properties of the Proof nets. This allows to have a good catch on the meaning of the ``paragraph'' modality, which is a peculiarity of light logics. On the other side, the proof of P-Time completeness, together with a lot of programming examples, gives a flavor of the non trivial task of programming with resource limitations, using Intuitionistic Light Affine Logic derivations as programs.